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\author{学号 \underline{\hspace{4cm}} \hspace{1cm} 姓名 \underline{\hspace{4cm}} }
\title{实变函数练习1.3-1.5}
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\date{2024 年 3 月 11 日}
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\begin{document}

\maketitle

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\begin{enumerate}

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\item  %Problem 01
构造下述集合之间的对等： 
\begin{enumerate}[label={(\arabic*)}]
\item  正奇数全体与正偶数全体。 
\item  正整数全体与正偶数全体。 
\item  区间 $(0,1)$ 与全体实数。 
\end{enumerate}


\vspace{0.1cm}

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\item  %Problem 02
证明对等关系是一种等价关系，即满足自反性、对称性和传递性。

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\item  %Problem 03
什么时候称两个集合有相同的基数？什么时候称一个集合的基数小于另一个集合的基数？

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\item  %Problem 04
证明伯恩斯坦定理：设 $A,B$ 是两个非空集合。如果 $A$ 对等于 $B$ 的一个子集，$B$ 也对等于 $A$ 的一个子集，那么 $A$ 与 $B$ 对等。

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\item  %Problem 05
证明：设 $A$ 是可数集，$B$ 是有限集或者可数集，则 $A\cup B$ 为可数集。

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\item  %Problem 06
证明：可数个可数集的并集仍是可数集。

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\item  %Problem 07
证明：有理数全体是一个可数集。

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\item  %Problem 08
证明：设 $A,B$ 都是可数集，则 $A\times B$ 也是可数集。

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\item  %Problem 09
设 $a<b$ 是两个实数。证明区间 $(a,b)$ 是一个不可数集合。

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\item  %Problem 10
设 $M$ 是一个集合，记 $P(M)$ 是 $M$ 的幂集合，即 $M$ 的所有子集组成的集合。证明：$P(M)$ 的基数大于 $M$ 的基数。


\vspace{0.1cm}


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\item[E1.]  %Problem 01
证明：
\begin{enumerate}[label={(\arabic*)}]
\item  $(A\backslash B)\backslash C = A\backslash (B\cup C)$. 
\item  $(A\cup B)\backslash C = (A\backslash C) \cup (A\backslash B)$. 
\end{enumerate}

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\item[E4.]  %Problem 02
设 $A_{2n-1}=(0,1/n), A_{2n}=(0,n), n=1,2,\cdots$. 求集合序列 $\{A_n\}$ 的上限集合与下限集合。

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\item[E7.]  %Problem 03
设 $f(x),g(x)$ 是定义在集合 $E$ 上的函数。证明：
$$\{x: f(x)>g(x) \} = \bigcup\limits_{n=1}^{\infty} \left\{ x: f(x) > g(x) + \frac{1}{n} \right\}. $$ 

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\item[E10.]  %Problem 04
设 $\{f_n(x)\}$ 是定义在 $E$ 上的一列函数，设 $c$ 是任意实数。证明：
$$\{x: \underset{n}\inf \{f_n(x)\} \ge c \} = \bigcap\limits_{n=1}^{\infty} \left\{ x: f_n(x) \ge c \right\}. $$ 

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\item[E14.]  %Problem 05
证明：球面上的点集与平面上的点集是对等的。

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\item[E17.]  %Problem 06
证明：增函数的不连续点最多只有可数个。

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\end{enumerate}


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\end{document}

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